Method based on stokes parameters for the adaptive adjustment of PMD compensators in optical fiber communication systems and compensator in accordance with said method

ABSTRACT

A method for the adaptive adjustment of a PMD compensator in optical fiber communication systems with the compensator comprising a cascade of adjustable optical devices through which passes an optical signal to be compensated and comprising the steps of computing the Stokes parameters S 0 , S 1 , S 2 , S 3  in a number Q of different frequencies of the signal output from the compensator, producing control signals for parameters of at least some of said adjustable optical devices so as to make virtually constant said Stokes parameters computed at different frequencies. A compensator comprising a cascade of adjustable optical devices ( 12 - 14 ) through which passes an optical signal to be compensated, an adjustment system which takes the components y 1 (t) e y 2 (t) on the two orthogonal polarizations from the signal at the compensator output, and which comprises a controller ( 15, 16 ) which on the basis of said components computes the Stokes parameters S 0 , S 1 , S 2 , S 3  in a number Q of different frequencies of the signal output by the compensator and which emits control signals for at least some of said adjustable optical devices so as to make virtually constant the Stokes parameters computed at the different frequencies.

The present invention relates to methods of adaptive adjustment of PMDcompensators in optical fiber communication systems. The presentinvention also relates to a compensator in accordance with said method.

In optical fiber telecommunications equipment the need to compensate theeffects of polarization mode dispersion (PMD) which occur when anoptical signal travels in an optical fiber based connection is known.

It is known that PMD causes distortion and dispersion of optical signalssent over optical fiber connections making the signals distorted anddispersed. The different time delays among the various signal componentsin the various polarization states acquire increasing importance withthe increase in transmission speeds. In modern optical fiber basedtransmission systems with ever higher frequencies (10 Gbit/s and more),accurate compensation of PMD effects becomes very important anddelicate. This compensation must be dynamic and performed at adequatespeed.

The general purpose of the present invention is to remedy the abovementioned shortcomings by making available a method of fast, accurateadaptive adjustment of a PMD compensator and a compensator in accordancewith said method.

In view of this purpose it was sought to provide in accordance with thepresent invention a method for the adaptive adjustment of a PMDcompensator in optical fiber communication systems with the compensatorcomprising a cascade of adjustable optical devices over which passes anoptical signal to be compensated comprising the steps of computing theStokes parameters S0, S1, S2, S3 in a number Q of different frequenciesof the signal output from the compensator, producing control signals forparameters of at least some of said adjustable optical devices so as tomake virtually constant said Stokes parameters computed at the differentfrequencies.

In accordance with the present invention it was also sought to realize aPMD compensator in optical fiber communication systems applying themethod and comprising a cascade of adjustable optical devices over whichpasses an optical signal to be compensated and an adjustment systemwhich takes the components y₁(t) and y₂(t) on the two orthogonalpolarizations at the compensator output with the adjustment systemcomprising a controller which on the basis of said components takencomputes the Stokes parameters S₀, S₁, S₂, S₃ in a number Q of differentfrequencies of the signal output from the compensator and which emitscontrol signals for at least some of said adjustable optical devices soas to make virtually constant the Stokes parameters computed at thedifferent frequencies.

To clarify the explanation of the innovative principles of the presentinvention and its advantages compared with the prior art there isdescribed below with the aid of the annexed drawings a possibleembodiment thereof by way of non-limiting example applying saidprinciples. In the drawings—

FIG. 1 shows a block diagram of a PMD compensator with associatedcontrol circuit, and

FIG. 2 shows an equivalent model of the PMD compensator.

With reference to the FIGS FIG. 1 shows the structure of a PMDcompensator designated as a whole by reference number 10. This structureconsists of the cascade of some optical devices which receive the signalfrom the transmission fiber 11. The first optical device is apolarization controller 12 (PC) which allows modification of the opticalsignal polarization at its input. There are three polarizationmaintaining fibers 13 (PMF) separated by two optical rotators 14.

A PMF fiber is a fiber which introduces a predetermined differentialunit delay (DGD) between the components of the optical signal on the twoprincipal states of polarization (PSP) termed slow PSP and fast PSP.

In the case of the compensator shown in FIG. 1 the DGD delays at thefrequency of the optical carrier introduced by the three PMFs arerespectively τ_(c), ατ_(c) and (1−α) τ_(c) with 0<α<1 and with τ_(c) andα which are design parameters.

An optical rotator is a device which can change the polarization of theoptical signal upon its input by an angle θ_(i) (the figure shows θ_(i)for the first rotator and θ₂ for the second) on a maximum circle on thePoincarè sphere.

An optical rotator is implemented in practice by means of a properlycontrolled PC.

In FIG. 1, x₁(t) and x₂(t) designate the components on the two PSPs ofthe optical signal at the compensator input whereas similarly y₁(t) andy₂(t) are the components of the optical signal at the compensatoroutput.

The input-output behavior of each optical device is described here bymeans of the so called Jones transfer matrix H(ω) which is a 2×2 matrixcharacterized by frequency dependent components. Designating by W₁(ω) eW₂(ω) the Fourier transforms of the optical signal components at thedevice input the Fourier transforms Z₁(ω) e Z₂(ω) of the optical signalcomponents at the device output are given by:

$\begin{matrix}{\begin{pmatrix}{Z_{1}(\omega)} \\{Z_{2}(\omega)}\end{pmatrix} = {{H(\omega)}\begin{pmatrix}{W_{1}(\omega)} \\{W_{2}(\omega)}\end{pmatrix}}} & (1)\end{matrix}$

Thus the Jones transfer matrix of the PC is:

$\begin{matrix}\begin{pmatrix}{h_{1}} & h_{2} \\{- h_{2}^{*}} & h_{1}^{*}\end{pmatrix} & (2)\end{matrix}$where h₁ e h₂ satisfy the condition |h₁|²+|h₂|²=1 and are frequencyindependent.

Denoting by φ₁ and φ₂ the PC control angles, h₁ and h₂ are expressed by:h ₁=−cos(φ₂−φ₁)+j sin(φ₂−φ₁)sin φ₁  (3)h ₂ =−j sin(φ₂−φ₁)cos φ₁

Clearly if the PC is controlled using other angles or voltages,different relationships will correlate these other parameters with h₁and h₂. The straightforward changes in the algorithms for adaptiveadjustment of the PMD compensator are discussed below.

Similarly, an optical rotator with rotation angle θ_(i) is characterizedby the following Jones matrix:

$\begin{matrix}\begin{pmatrix}{\cos\;\theta_{i}} & {\sin\;\theta_{i}} \\{{- \sin}\;\theta_{i}} & {\cos\;\theta_{i}}\end{pmatrix} & (4)\end{matrix}$

The Jones transfer matrix of a PMF with DGD τ_(i) may be expressed asRDR⁻¹ where D is defined as:

$\begin{matrix}{D\hat{=}\begin{pmatrix}{\mathbb{e}}^{{j\omega\tau}_{i}/2} & 0 \\0 & {\mathbb{e}}^{{- {j\omega\tau}_{i}}/2}\end{pmatrix}} & (5)\end{matrix}$and R is a unitary rotation matrix accounting for the PSPs' orientation.This matrix R may be taken as the identity matrix I without loss ofgenerality when the PSPs of all the PMFs are aligned.

As shown in FIG. 1, to control the PMD compensator a controller 15 isneeded to produce optical device control signals of the compensatorcomputed on the basis of the quantities sent to it by a controller pilot16 termed controller driver (CD).

The CD feeds the controller with the quantities needed to update thecompensator optical device control parameters. As described below, thesequantities will be extracted by the CD from the signals at the inputand/or output of the compensator.

The controller will operate following the criterion described below andwill use one of the two algorithms described below.

To illustrate the PMD compensator adaptive adjustment algorithms let usassume that the controller can directly control the parameters φ₁, φ₂,θ₁ and θ₂ which we consolidate in a vector θ defined as:θ{circumflex over (=)}(φ₁, φ₂, θ₁, θ₂)^(T)

If it is not so, in general there will be other parameters to control,for example some voltages, which will be linked to the previous ones inknown relationships.

The time instants in which the update of the compensator parameters isrealized are designated t_(n) (con n=0, 1, 2 . . . , ), and T_(u)designates the time interval between two successive updates, thust_(n+1)=t_(n)+T_(u). In addition, θ(t_(n)) designates the value of thecompensator parameters after the nth update.

In accordance with the method of the present invention the criterion foradjusting the compensator parameters employs the so-called Stokesparameters. Computation of the Stokes parameters for an optical signalis well known to those skilled in the art and is not further described.

Again in accordance with the method the parameters θ of the compensatorare adjusted to make constant the Stokes parameters computed atdifferent frequencies on the compensator output signal. The four Stokesparameters S₀, S₁, S₂ e S₃ computed at the frequency f_(l) aredesignated by:S ₀|_(f=f) _(l) {circumflex over (=)}S _(0,l)S ₁|_(f=f) _(l) {circumflex over (=)}S _(1,l)S ₂|_(f=f) _(l) {circumflex over (=)}S _(2,l)S ₃|_(f=f) _(l) {circumflex over (=)}S _(3,l)

Similarly, the Stokes parameters computed at the frequency f_(p) aredesignated by S_(0,p), S_(1,p), S_(2,p) e S_(3,p).

Using these Stokes parameters the following unitary vectors areconstructed with components given by the three Stokes parameters S₁, S₂,S₃ normalized at the parameter S₀. (.)^(T) below designates thetranspose while (.)* designates the complex conjugate:

$( {\frac{S_{1,l}}{S_{0,l}},\frac{S_{2,l}}{S_{0,l}},\frac{S_{3,l}}{S_{0,l}},} )^{T}\mspace{14mu}{and}\mspace{14mu}( {\frac{S_{1,p}}{S_{0,p}},\frac{S_{2,p}}{S_{0,p}},\frac{S_{3,p}}{S_{0,p}},} )^{T}$

In the absence of PMD these two vectors are parallel. Consequently, iftheir quadratic Euclidean distance is considered G_(1p)(θ):

$\begin{matrix}{{G_{lp}(\theta)} = {( {\frac{S_{1,l}}{S_{0,l}} - \frac{S_{1,p}}{S_{0,p}}} )^{2} + ( {\frac{S_{2,l}}{S_{0,l}} - \frac{S_{2,p}}{S_{0,p}}} )^{2} + ( {\frac{S_{3,l}}{S_{0,l}} - \frac{S_{3,p}}{S_{0,p}}} )^{2}}} & (6)\end{matrix}$which is a function of the parameters θ of the PMD compensator it willbe zero when the PMD is compensated at the two frequencies consideredf_(l) and f_(p).

Now consider a number Q of frequencies f_(l), l=1, 2, . . . , Q. Computethe Stokes parameters at these frequencies and construct thecorresponding units defined as explained above, i.e. with componentsgiven by the three Stokes parameters S₁, S₂, S₃ normalized with respectto the parameter S₀. All these units are parallel if and only if the sumof their quadratic Euclidean distances is zero.

Consequently, to adaptively adjust the PMD compensator parameters wedefine the function G(θ) which is to be minimized as the sum of thequadratic distances G_(1p)(θ) with 1,p=1, 2, . . . , Q, i.e. the sum ofthe quadratic distances of the pair of vectors at the differentfrequencies f_(l) and f_(p), for l,p=1, 2, . . . Q:

$\begin{matrix}{{G(\theta)}\hat{=}{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}{G_{lp}(\theta)}}}} & (7)\end{matrix}$

The update rule for the compensator parameters to be used in accordancewith the present invention are:

$\begin{matrix}{\begin{matrix}{{{\phi_{1}( t_{n + 1} )} = {{\phi_{1}( t_{n} )} - {\gamma\frac{\partial{G(\theta)}}{\partial\phi_{1}}}}}}_{\theta = {\theta{(t_{n})}}} \\{{= {{\phi_{1}( t_{n} )} - {\gamma{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}\frac{\partial{G_{lp}(\theta)}}{\partial\phi_{1}}}}}}}}_{\theta = {\theta{(t_{n})}}}\end{matrix}\begin{matrix}{{\phi_{2}( t_{n + 1} )} = {{{\phi_{2}( t_{n} )} - {\gamma\frac{\partial{G(\theta)}}{\partial\phi_{2}}}}❘_{\theta = {\theta{(t_{n})}}}}} \\{{= {{\phi_{2}( t_{n} )} - {\gamma{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}\frac{\partial{G_{lp}(\theta)}}{\partial\phi_{2}}}}}}}}_{\theta = {\theta{(t_{n})}}}\end{matrix}\begin{matrix}{{{\theta_{1}( t_{n + 1} )} = {{\theta_{1}( t_{n} )} - {\gamma\frac{\partial{G(\theta)}}{\partial\theta_{1}}}}}}_{\theta = {\theta{(t_{n})}}} \\{{= {{\phi_{1}( t_{n} )} - {\gamma{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}\frac{\partial{G_{lp}(\theta)}}{\partial\theta_{1}}}}}}}}_{\theta = {\theta{(t_{n})}}}\end{matrix}\begin{matrix}{{\theta_{2}( t_{n + 1} )} = {{{\theta_{2}( t_{n} )} - {\gamma\frac{\partial{G(\theta)}}{\partial\theta_{2}}}}❘_{\theta = {\theta{(t_{n})}}}}} \\{= {{{\theta_{2}( t_{n} )} - {\gamma{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}\frac{\partial{G_{lp}(\theta)}}{\partial\theta_{2}}}}}}❘_{\theta = {\theta{(t_{n})}}}}}\end{matrix}} & (8)\end{matrix}$where γ>0 is a scale factor which controls the amount of the adjustment.

In vector notation this means that the vector of the compensatorparameters is updated by adding a new vector with its norm proportionateto the norm of the gradient of G(θ) and with opposite direction, i.e.with all its components having their sign changed. This way, we are sureto move towards a relative minimum of the function G(θ).

All this is equivalent to:

$\begin{matrix}{{{{\theta( t_{n + 1} )} = {{\theta( t_{n} )} - {\gamma{\nabla{G(\theta)}}}}}}_{\theta = {\theta{(t_{n})}}}\mspace{70mu} = {{{\theta( t_{n} )} - {\gamma{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}{\nabla{G_{lp}(\theta)}}}}}}❘_{\theta = {\theta{(t_{n})}}}}} & (9)\end{matrix}$

A simplified version of (9) consists of an update by means of a constantnorm vector and therefore an update which uses only the information onthe direction of ∇G(θ). In this case the update rule becomes.

$\begin{matrix}{{\theta( t_{n + 1} )} = {{{{\theta( t_{n} )} - {{\gamma sign}{\nabla{G(\theta)}}}}❘_{\theta = {\theta{(t_{n})}}}}\mspace{70mu} = {{{\theta( t_{n} )} - {{\gamma sign}{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}{\nabla{G_{lp}(\theta)}}}}}}❘_{\theta = {\theta{(t_{n})}}}}}} & (10)\end{matrix}$where sign (z) designates a vector with unitary components and of thesame sign as the components or the vector z.

Two methods are now described for computing the gradient of the G(θ)function and obtaining the required control parameters.

First Method

To implement the update rules (8) the partial derivatives of G(θ) forθ=θ(t_(n)) can be computed using the following five-step procedure.

-   -   Step 1. find the value of G[θ(t_(n))]=G[φ₁(t_(n)), φ₂(t_(n)),        θ₁(t_(n)), θ₂(t_(n))] at iteration n. To do this, in the time        interval (t_(n), t_(n)+T_(u)/5) the Stokes parameters at the        above mentioned Q frequencies are derived and the value of the        function G(θ) is computed using equations (6) and (7).    -   Step 2. find the partial derivative

${\frac{\partial{G(\theta)}}{\partial\phi_{1}}}_{\theta = {\theta{(t_{n})}}}$

-   -    at iteration n. To do this, parameter φ₁ is set at φ₁(t_(n))+Δ        while the other parameters are left unchanged. The corresponding        value of G(θ), i.e. G[φ₁(t_(n))+Δ, φ₂(t_(n)), θ₁(t_(n)),        θ₂(t_(n))], is computed as in step 1 but in the time interval        (t_(n)+T_(u)/5, t_(n)+2T_(u)/5). The estimate of the partial        derivative of G(θ) as a function of φ₁ is computed as:

$\begin{matrix}{{\frac{\partial{G(\theta)}}{\partial\phi_{1}}}_{\theta = {\theta{(t_{n})}}} \cong \frac{\begin{matrix}{{G\lbrack {{{\phi_{1}( t_{n} )} + \Delta},{\phi_{2}( t_{n} )},{\theta_{1}( t_{n} )},{\theta_{2}( t_{n} )}} \rbrack} -} \\{G\lbrack {{\phi_{1}( t_{n} )},{\phi_{2}( t_{n} )},{\theta_{1}( t_{n} )},{\theta_{2}( t_{n} )}} \rbrack}\end{matrix}}{\Delta}} & (11)\end{matrix}$

-   -   Step 3. Find the partial derivative:

${\frac{\partial{G(\theta)}}{\partial\phi_{2}}}_{\theta = {\theta{(t_{n})}}}$

-   -    at iteration n. To do this the parameter φ₂ is set at        φ₂(t_(n))+Δ while the other parameters are left changed. The        corresponding value of G(θ), i.e. G[φ₁(t_(n)), φ₂(t_(n))+Δ,        θ₁(t_(n)), θ₂(t_(n))], )], is computed as in step 1 but in the        time interval (t_(n)2T_(u)/5, t_(n)+3T_(u)/5). The estimate of        the partial derivative of G(θ) with respect to φ₂ is computed        as:

$\begin{matrix}{{\frac{\partial{G(\theta)}}{\partial\phi_{2}}}_{\theta = {\theta{(t_{n})}}} \cong \frac{\begin{matrix}{{G\lbrack {{\phi_{1}( t_{n} )},{{\phi_{2}( t_{n} )} + \Delta},{\theta_{1}( t_{n} )},{\theta_{2}( t_{n} )}} \rbrack} -} \\{G\lbrack {{\phi_{1}( t_{n} )},{\phi_{2}( t_{n} )},{\theta_{1}( t_{n} )},{\theta_{2}( t_{n} )}} \rbrack}\end{matrix}}{\Delta}} & (12)\end{matrix}$

-   -   Step 4: Find the partial derivative:

${\frac{\partial{G(\theta)}}{\partial\theta_{1}}}_{\theta = {\theta{(t_{n})}}}$

-   -    at iteration n. To do this, parameter θ₁ is set at θ₁(t_(n))+Δ        while the other parameters are left unchanged, the corresponding        value of G(θ), i.e. G[φ₁(t_(n)), φ₂(t_(n)), θ₁(t_(n))+Δ,        θ₂(t_(n))], is computed as in Step 1 but in the time interval        (t_(n)+3T_(u)/5, t_(n)+4T_(u)/5) and the estimate of the partial        derivative of G(θ) with respect to θ₁ is computed as:

$\begin{matrix}{{\frac{\partial{G(\theta)}}{\partial\theta_{1}}}_{\theta = {\theta{(t_{n})}}} \cong \frac{\begin{matrix}{{G\lbrack {{\phi_{1}( t_{n} )},{\phi_{2}( t_{n} )},{{\theta_{1}( t_{n} )} + \Delta},{\theta_{2}( t_{n} )}} \rbrack} -} \\{G\lbrack {{\phi_{1}( t_{n} )},{\phi_{2}( t_{n} )},{\theta_{1}( t_{n} )},{\theta_{2}( t_{n} )}} \rbrack}\end{matrix}}{\Delta}} & (13)\end{matrix}$

-   -   Step 5: Find the partial derivative:

${\frac{\partial{G(\theta)}}{\partial\theta_{2}}}_{\theta = {\theta{(t_{n})}}}$

-   -    at iteration n. To do this the parameter φ₂ is set at        φ₂(t_(n))+Δ while the other parameters are left changed. The        corresponding value of G(θ), i.e. G[φ₁(t_(n)), φ₂(t_(n)),        θ₁(t_(n)), θ₂(t_(n))+Δ], is computed as in step 1 but in the        time interval (t_(n)+4T_(u)/5, t_(n)+T_(u)). The estimate of the        partial derivative of G(θ) with respect to φ₂ is computed as:

$\begin{matrix}{{\frac{\partial{G(\theta)}}{\partial\theta_{2}}}_{\theta = {\theta{(t_{n})}}} \cong \frac{\begin{matrix}{{G\lbrack {{\phi_{1}( t_{n} )},{\phi_{2}( t_{n} )},{\theta_{1}( t_{n} )},{{\theta_{2}( t_{n} )} + \Delta}} \rbrack} -} \\{G\lbrack {{\phi_{1}( t_{n} )},{\phi_{2}( t_{n} )},{\theta_{1}( t_{n} )},{\theta_{2}( t_{n} )}} \rbrack}\end{matrix}}{\Delta}} & (14)\end{matrix}$

The above parameter update is done only after estimation of the gradienthas been completed.

Note that in this case it is not necessary that the relationship betweenthe control parameters of PC and optical rotators and the correspondingJones matrices be known.

Indeed, the partial derivatives of the function with respect to thecompensator control parameters are computed without knowledge of thisrelationship. Consequently if the control parameters are different fromthose assumed as an example and are for example some voltage or someother angle, we may similarly compute the partial derivative and updatethese different control parameters accordingly.

Lastly, it is noted that when this algorithm is used the CD must receiveonly the optical signal at the compensator output and must supply thecontroller with the Stokes parameters computed at the Q frequenciesf_(l), l=1, 2, . . . , Q.

Second Method

When an accurate characterization of the PC and of each optical rotatoris available the update rules can be expressed as a function of thesignals on the two orthogonal polarizations at the compensator input andoutput.

In this case, for the sake of convenience it is best to avoidnormalization of the three Stokes parameters S₁, S₂ e S₃ with respect toS₀ and use the function H(θ) defined as:

$\begin{matrix}{{{H(\theta)} = {\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}{H_{lp}(\theta)}}}}{where}} & (15) \\{{H_{lp}(\theta)} = {( {S_{1,l} - S_{1,p}} )^{2} + ( {S_{2,1} - S_{2,p}} )^{2} + ( {S_{3,1} - S_{3,p}} )^{2}}} & (16)\end{matrix}$

Consequently we have new update rules similar to those expressed byequation (8) or equivalently (9) with the only change being that the newfunction H(θ) must substitute the previous G(θ).

Before describing how the gradient of this new function H(θ) is to becomputed let us introduce for convenient an equivalent model of the PMDcompensator.

Indeed it was found that the PMD compensator shown in FIG. 1 isequivalent to a two-dimensional transversal filter with four tappeddelay lines (TDL) combining the signals on the two principalpolarization states (PSP). This equivalent model is shown in FIG. 2where:c ₁{circumflex over (=)}cos θ₁ cos θ₂ h ₁c ₂{circumflex over (=)}−sin θ₁ sin θ₂ h ₁c ₃{circumflex over (=)}−sin θ₁ cos θ₂ h ₂*c ₄{circumflex over (=)}−cos θ₁ sin θ₂ h ₂*c ₅{circumflex over (=)}cos θ₁ cos θ₂ h ₂c ₆{circumflex over (=)}−sin θ₁ sin θ₂ h ₂c ₇{circumflex over (=)}sin θ₁ cos θ₂ h ₁*c ₈{circumflex over (=)}cos θ₁ sin θ₂ h ₁*  (18)

For the sake of convenience let c(θ)designate the vector whosecomponents are the c₁ in (17). It is noted that the tap coefficientsc_(i) of the four TDLs are not independent of each other. On thecontrary, given four of them the others are completely determined by(17). In the FIG for the sake of clarity it is designated β=1−α.

The gradient of H_(1p)(θ) with respect to θ is to be computed asfollows:

$\begin{matrix}{{\nabla{H_{lp}(\theta)}} = {4( {S_{1,l} - S_{1,p}} ){Re}\{ {\frac{1}{T_{u}}{\int_{t_{n}}^{t_{n + 1}}\lbrack {{{y_{1,l}^{*}(t)}{a_{l}^{T}(t)}} - {{y_{2,l}(t)}{b_{l}^{T}(t)}} -} }} }} \\{ { {{{y_{1,p}^{*}(t)}{a_{p}^{T}(t)}} + {{y_{2,p}^{*}(t)}{b_{p}^{T}(t)}}} \rbrack{\mathbb{d}{tJ}}} \} +} \\{4( {S_{2,l} - S_{2,p}} ){Re}\{ {\frac{1}{T_{u}}{\int_{t_{n}}^{t_{n + 1}}\lbrack {{{y_{2,l}^{*}(t)}{a_{l}^{T}(t)}} + {{y_{1,l}(t)}{b_{l}^{T}(t)}} -} }} } \\{ { {{{y_{2,p}^{*}(t)}{a_{p}^{T}(t)}} - {{y_{1,p}^{*}(t)}{b_{j}^{T}(t)}}} \rbrack{\mathbb{d}{tJ}}} \} -} \\{4( {S_{3,l} - S_{3,p}} ){Im}\{ {\frac{1}{T_{u}}{\int_{t_{n}}^{t_{n + 1}}\lbrack {{{y_{2,l}^{*}(t)}{a_{l}^{T}(t)}} + {{y_{1,l}(t)}{b_{l}^{T}(t)}} -} }} } \\ { {{{y_{2,p}^{*}(t)}{a_{p}^{T}(t)}} - {{y_{1,p}^{*}(t)}{b_{p}^{T}(t)}}} \rbrack{\mathbb{d}{tJ}}} \}\end{matrix}$where:

-   -   y_(1,l)(t) and y_(2,l)(t) are the signals y₁(t) and y₂(t) at the        compensator output respectively filtered through a narrow band        filter centered on the frequency f_(l) (similarly for y_(1,p)(t)        and y_(2,p)(t));    -   a_(l)(t) and b_(l)(t) are the vectors:

$\begin{matrix}{{a_{l}(t)} = \begin{pmatrix}{x_{1,l}(t)} \\{x_{1,l}( {t - {\alpha\tau}_{c}} )} \\{x_{1,l}( {t - \tau_{c}} )} \\{x_{1,l}( {t - \tau_{c} - {\alpha\tau}_{c}} )} \\{x_{2,l}(t)} \\{x_{2,l}( {t - {\alpha\tau}_{c}} )} \\{x_{2,l}( {t - \tau_{c}} )} \\{x_{2,l}( {t - \tau_{c} - {\alpha\tau}_{c}} )}\end{pmatrix}} & {{b_{l}(t)} = \begin{pmatrix}{x_{2,l}^{*}( {t - {2\tau_{c}}} )} \\{x_{2,l}^{*}( {t - \tau_{c} - {\beta\tau}_{c}} )} \\{x_{2,l}^{*}( {t - \tau_{c}} )} \\{x_{2,l}^{*}( {t - {\beta\tau}_{c}} )} \\{- {x_{1,l}^{*}( {t - {2\tau_{c}}} )}} \\{- {x_{1,l}^{*}( {t - \tau_{c} - {\beta\tau}_{c}} )}} \\{- {x_{1,l}^{*}( {t - \tau_{c}} )}} \\{- {x_{1,l}^{*}( {t - {\beta\tau}_{c}} )}}\end{pmatrix}}\end{matrix}$

-   -    with x_(1,l)(t) and x_(2,l)(t) which are respectively the        signals x₁(t) and x₂(t) at the compensator input filtered by a        narrow band filter centered on the frequency f_(l) (similarly        for y_(1,p)(t) and y_(2,p)(t));    -   J is the Jacobean matrix of the transformation c=c(θ) defined as

$\begin{matrix}{J\hat{=}\begin{pmatrix}\frac{\partial c_{1}}{\partial\phi_{1}} & \frac{\partial c_{1}}{\partial\phi_{2}} & \frac{\partial c_{1}}{\partial\theta_{1}} & \frac{\partial c_{1}}{\partial\theta_{2}} \\\frac{\partial c_{2}}{\partial\phi_{1}} & \frac{\partial c_{2}}{\partial\phi_{2}} & \frac{\partial c_{2}}{\partial\theta_{1}} & \frac{\partial c_{2}}{\partial\theta_{2}} \\\vdots & \vdots & \vdots & \vdots \\\frac{\partial c_{8}}{\partial\phi_{1}} & \frac{\partial c_{8}}{\partial\phi_{2}} & \frac{\partial c_{8}}{\partial\theta_{1}} & \frac{\partial c_{8}}{\partial\theta_{2}}\end{pmatrix}} & (18)\end{matrix}$

The parameters θ are updated in accordance with the rule

$\begin{matrix}{{{\theta( t_{n + 1} )} = {{\theta( t_{n} )} - {\gamma{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}{\nabla{H_{lp}(\theta)}}}}}}}}_{\theta = {\theta{(t_{n})}}} & (19)\end{matrix}$or in accordance with the following simplified rule based only on thesign:

$\begin{matrix}{{{\theta( t_{n + 1} )} = {{\theta( t_{n} )}{\gamma sign}{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l1}{\nabla{H_{lp}(\theta)}}}}}}}_{\theta = {\theta{(t_{n})}}} & (20)\end{matrix}$

When the control parameters are different from those taken as exampleswe will naturally have different relationships between these controlparameters and the coefficients c_(i).

For example, if the PC is controlled by means of some voltages, giventhe relationship between these voltages and the coefficients h₁ and h₂which appear in (2), by using the equations (17) we will be able toexpress the coefficients c_(i) as a function of these new controlparameters.

Consequently in computing the gradient of the function H(θ), the onlychange we have to allow for is the expression of the Jacobean matrix J,which has to be changed accordingly.

Lastly it is noted that when this second method is used the CD mustreceive the optical signals at the input and output of the compensator.The CD must supply the controller not only with the Stokes parametersfor the optical signal at the compensator output and computed at the Qfrequencies f_(l), l=1, 2, . . . , Q but also with the signalsx_(1,l)(t), x_(2,l)(t), y_(1,l)(t) e y_(2,l)(t) corresponding to the Qfrequencies f_(l), l=1, 2, . . . , Q.

It is now clear that the predetermined purposes have been achieved bymaking available an effective method for adaptive control of a PMDcompensator and a compensator applying this method.

Naturally the above description of an embodiment applying the innovativeprinciples of the present invention is given by way of non-limitingexample of said principles within the scope of the exclusive rightclaimed here.

1. Method for the adaptive adjustment of a PMD compensator in opticalfiber communication systems with the compensator comprising a cascade ofadjustable optical devices over which passes an optical signal to becompensated comprising the steps of: computing the Stokes parameters S₀,S₁, S₂, S₃ in a number Q of different frequencies of the compensatoroutput signal, and producing control signals for parameters of at leastsome of said adjustable optical devices so as to make virtually constantsaid Stokes parameters computed at different frequencies.
 2. Method inaccordance with claim 1 comprising the steps of computing the Stokesparameters in pairs of frequencies fl and fp with l,p=1, 2, . . , Q,obtaining at the lth and pth frequencies of the Q frequencies the twoseries of Stokes parameters S_(0,l), S_(1,l), S_(2,l), S_(3,l) andS_(0,p), S_(1,p), S_(2,p), S_(3,p), computing a vector function of eachseries of Stokes parameters and producing the control signals in such amanner that said vectors function of the two series of parameters arevirtually parallel to each other.
 3. Method in accordance with claim 2in which said vectors are unitary norm vectors with components given bythe Stokes parameters S₁, S₂, S₃ normalized to the Stokes parameter S₀,i.e.:$( {\frac{S_{1,l}}{S_{0,l}},\frac{S_{2,l}}{S_{0,l}},\frac{S_{3,l}}{S_{0,l}},} )^{T}$${{and}( {\frac{S_{1,p}}{S_{0,p}},\frac{S_{2,p}}{S_{0,p}},\frac{S_{3,p}}{S_{0,p}},} )}^{T}.$4. Method in accordance with claim 3 in which is defined the function:${G(\theta)}\hat{=}{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}{G_{lp}(\theta)}}}$${{with}\mspace{14mu}{G_{lp}(\theta)}} = {( {\frac{S_{1,l}}{S_{0,l}} - \frac{S_{1,p}}{S_{0,p}}} )^{2} + ( {\frac{S_{2,l}}{S_{0,l}} - \frac{S_{2,p}}{S_{0,p}}} )^{2} + ( {\frac{S_{3,l}}{S_{0,l}} - \frac{S_{3,p}}{S_{0,p}}} )^{2}}$and the control signals are produced to minimize said function G(θ). 5.Method in accordance with claim 4 in which the optical devices comprisea polarization controller with controllable angles φ₁, φ₂ and tworotators with controllable rotation angles respectively θ₁ and θ₂, andto minimize the function G(θ) the updating of φ₁, φ₂, θ₁ and θ₂ of thecompensator observes the following rules to go from the nth iteration tothe n+1th iteration: $\begin{matrix}{{{\phi_{1}( t_{n + 1} )} = {{\phi_{1}( t_{n} )} - {\gamma\frac{\partial{G(\theta)}}{\partial\phi_{1}}}}}}_{\theta = {\theta{(t_{n})}}} \\{=  {{\phi_{1}( t_{n} )} - {\gamma{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}\frac{\partial{G_{lp}(\theta)}}{\partial\phi_{1}}}}}} |_{\theta = {\theta{(t_{n})}}}}\end{matrix}$ $\begin{matrix}{{{\phi_{2}( t_{n + 1} )} = {{\phi_{2}( t_{n} )} - {\gamma\frac{\partial{G(\theta)}}{\partial\phi_{2}}}}}}_{\theta = {\theta{(t_{n})}}} \\{=  {{\phi_{2}( t_{n} )} - {\gamma{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}\frac{\partial{G_{lp}(\theta)}}{\partial\phi_{2}}}}}} |_{\theta = {\theta{(t_{n})}}}}\end{matrix}$ $\begin{matrix}{{{\theta_{1}( t_{n + 1} )} = {{\theta_{1}( t_{n} )} - {\gamma\frac{\partial{G(\theta)}}{\partial\theta_{1}}}}}}_{\theta = {\theta{(t_{n})}}} \\{=  {{\theta_{1}( t_{n} )} - {\gamma{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}\frac{\partial{G_{lp}(\theta)}}{\partial\theta_{1}}}}}} |_{\theta = {\theta{(t_{n})}}}}\end{matrix}$ $\begin{matrix}{{{\theta_{2}( t_{n + 1} )} = {{\theta_{2}( t_{n} )} - {\gamma\frac{\partial{G(\theta)}}{\partial\theta_{2}}}}}}_{\theta = {\theta{(t_{n})}}} \\{=  {{\theta_{2}( t_{n} )} - {\gamma{\sum\limits_{l = 2}^{Q}{\sum\limits_{p = 1}^{l - 1}\frac{\partial{G_{lp}(\theta)}}{\partial\theta_{2}}}}}} \middle| {}_{\theta = {\theta{(t_{n})}}}. }\end{matrix}$
 6. Method in accordance with claim 5 in which the partialderivatives of G(θ) for θ=θ(t_(n)) are computed in accordance with thefollowing steps: Step
 1. find the value of G[θ(t_(n))]=G[φ₁(t_(n)),φ₂(t_(n)), θ₁(t_(n)), θ₂(t_(n))] at iteration n; to do this, in the timeinterval (t_(n), t_(n)+T_(u)/5) the Stokes parameters at the Qfrequencies are derived and the value of the function G(θ) is computedStep
 2. find the partial derivative${\frac{\partial{G(\theta)}}{\partial\phi_{1}}}_{\theta = {\theta{(t_{n})}}}$ at iteration n; to do this, parameter φ₁ is set at φ₁(t_(n))+Δ whilethe other parameters are left unchanged, the corresponding value ofG(θ), i.e. G[φ₁(t_(n))+Δ, φ₂(t_(n)), θ₁(t_(n)), θ₂(t_(n))], is computedas in step 1 but in the time interval (t_(n)+T_(u)/5, t_(n)+2T_(u)/5)and the estimate of the partial derivative of G(θ) with respect to φ₁ iscomputed as:${\frac{\partial{G(\theta)}}{\partial\phi_{1}}}_{\theta = {\theta{(t_{n})}}} \cong \frac{\begin{matrix}{{G\lbrack {{{\phi_{1}( t_{n} )} + \Delta},{\phi_{2}( t_{n} )},{\theta_{1}( t_{n} )},{\theta_{2}( t_{n} )}} \rbrack} -} \\{G\lbrack {{\phi_{1}( t_{n} )},{\phi_{2}( t_{n} )},{\theta_{1}( t_{n} )},{\theta_{2}( t_{n} )}} \rbrack}\end{matrix}}{\Delta}$ Step
 3. Find the partial derivative:${\frac{\partial{G(\theta)}}{\partial\phi_{2}}}_{\theta = {\theta{(t_{n})}}}$ at iteration n; to do this the parameter φ₂ is set at φ₂(t_(n))+Δ whilethe other parameters are left changed, the corresponding value of G(θ),i.e. G[φ₁(t_(n)), φ₂(t_(n))+Δ, θ₁(t_(n)), θ₂(t_(n))], is computed as instep 1 but in the time interval (t_(n)+2T_(u)/5, t_(n)+3T_(u)/5) and theestimate of the partial derivative of G(θ) with respect to φ₂ iscomputed as:$ \frac{\partial{G(\theta)}}{\partial\phi_{2}} \middle| {}_{\theta = {\theta{(t_{n})}}}{\cong \frac{\begin{matrix}{{G\lbrack {{\phi_{1}( t_{n} )},{{\phi_{2}( t_{n} )} + \Delta},{\theta_{1}( t_{n} )},{\theta_{2}( t_{n} )}} \rbrack} -} \\{G\lbrack {{\phi_{1}( t_{n} )},{\phi_{2}( t_{n} )},{\theta_{1}( t_{n} )},{\theta_{2}( t_{n} )}} \rbrack}\end{matrix}}{\Delta}} $ Step
 4. Find the partial derivative:$ \frac{\partial{G(\theta)}}{\partial\theta_{1}} |_{\theta = {\theta{(t_{n})}}}$ at iteration n; to do this, parameter θ₁ is set at θ₁(t_(n))+Δ whilethe other parameters are left unchanged, the corresponding value ofG(θ), i.e. G[φ₁(t_(n)) φ₂(t_(n)), θ₁(t_(n))+Δ, θ₂(t_(n))], is computedas in Step 1 but in the time interval (t_(n)+3T_(u)/5, t_(n)+4T_(u)/5)and the estimate of the partial derivative of G(θ) with respect to θ₁ iscomputed as:$ \frac{\partial{G(\theta)}}{\partial\theta_{1}} \middle| {}_{\theta = {\theta{(t_{n})}}}{\cong \frac{\begin{matrix}{{G\lbrack {{\phi_{1}( t_{n} )},{\phi_{2}( t_{n} )},{{\theta_{1}( t_{n} )} + \Delta},{\theta_{2}( t_{n} )}} \rbrack} -} \\{G\lbrack {{\phi_{1}( t_{n} )},{\phi_{2}( t_{n} )},{\theta_{1}( t_{n} )},{\theta_{2}( t_{n} )}} \rbrack}\end{matrix}}{\Delta}} $ Step
 5. Find the partial derivative:$ \frac{\partial{G(\theta)}}{\partial\theta_{2}} |_{\theta = {\theta{(t_{n})}}}$ at iteration n; to do this the parameter φ₂ is set at φ₂(t_(n))+Δ whilethe other parameters are left unchanged, the corresponding value ofG(θ), i.e. G[φ₁(t_(n)), φ₂(t_(n)), θ₁(t_(n)), θ₂(t_(n))+Δ], is computedas in step 1 but in the time interval (t_(n)+4T_(u)/5, t_(n)+T_(u)) andthe estimate of the partial derivative of G(θ) with respect to φ₂ iscomputed as:$ \frac{\partial{G(\theta)}}{\partial\theta_{2}} \middle| {}_{\theta = {\theta{(t_{n})}}}{\cong {\frac{\begin{matrix}{{G\lbrack {{\phi_{1}( t_{n} )},{\phi_{2}( t_{n} )},{\theta_{1}( t_{n} )},{{\theta_{2}( t_{n} )} + \Delta}} \rbrack} -} \\{G\lbrack {{\phi_{1}( t_{n} )},{\phi_{2}( t_{n} )},{\theta_{1}( t_{n} )},{\theta_{2}( t_{n} )}} \rbrack}\end{matrix}}{\Delta}.}} $
 7. Method in accordance with claim 1comprising the steps of computing the Stokes parameters in pairs offrequencies f_(l) and f_(p) with l,p=1, 2, . . . , Q, to obtain at thelth and pth frequencies of the Q frequencies the two series of Stokesparameters S_(1,l), S_(2,l), S_(3,l) e S_(1,p), S_(2,p), S_(3,p),defining the function: $\begin{matrix}{{H(\theta)} = {\sum\limits_{l = 2}^{Q}\;{\sum\limits_{p = 1}^{l - 1}\;{H_{lp}(\theta)}}}} \\{{{with}\mspace{14mu}{H_{lp}(\theta)}} = {( {S_{1,l} - S_{1,p}} )^{2} + ( {S_{2,l} - S_{2,p}} )^{2} + ( {S_{3,l} - S_{3,p}} )^{2}}}\end{matrix}$ with and producing said control signals to minimize saidfunction H(θ).
 8. Method in accordance with claim 7 in which the opticaldevices comprise a polarization controller with controllable angles φ₁,φ₂ and two rotators with controllable rotation angles respectively θ₁and θ₂, and for minimizing the function H(θ) the updating of φ₁, φ₂, θ₁and θ₂, of the compensator follows the following mles for passing fromthe nth iteration to the n+1th e iteration:${\theta( t_{n + 1} )} =  {{\theta( t_{n} )} - {\gamma{\sum\limits_{l = 2}^{Q}\;{\sum\limits_{p = 1}^{l - 1}\;{\nabla{H_{lp}(\theta)}}}}}} |_{\theta = {\theta{(t_{n})}}}$or the following simplified rule:${\theta( t_{n + 1} )} = {{\theta( t_{n} )} - {{\gamma sign}\lbrack  {\sum\limits_{l = 2}^{Q}\;{\sum\limits_{p = 1}^{l - 1}\;{\nabla{H_{lp}(\theta)}}}} |_{\theta = {\theta{(t_{n})}}} \rbrack}}$with ∇H_(LP)(θ) equal to the gradient of H_(1p)(θ) with respect to{tilde over (θ)}.
 9. Method in accordance with claim 7 in which saidparameters are consolidated in a vector θ which is updated in accordancewith the rule $\begin{matrix}{{\theta( t_{n + 1} )} =  {{\theta( t_{n} )} - {\gamma{\sum\limits_{l = 2}^{Q}\;{\sum\limits_{p = 1}^{l - 1}\;{\nabla{H_{lp}(\theta)}}}}}} |_{\theta = {\theta{(t_{n})}}}} & (19)\end{matrix}$ or the following simplified rule based only on the sign:$\begin{matrix}{{\theta( t_{n + 1} )} = {{\theta( t_{n} )} - {{\gamma sign}\lbrack  {\sum\limits_{l = 2}^{Q}\;{\sum\limits_{p = 1}^{l - 1}\;{\nabla{H_{lp}(\theta)}}}} |_{\theta = {\theta{(t_{n})}}} \rbrack}}} & (20)\end{matrix}$ with ∇H_(LP)(θ) equal to the gradient of H_(lp)(θ) withrespect to {tilde over (θ)}.
 10. Method in accordance with claim 9 inwhich between the controller and an optical rotator and between opticalrotators there are fibers which introduce a predetermined differentialunit delay maintaining the polarization.
 11. Method in accordance withclaim 1 in which the PMD compensator is modeled like a two-dimensionaltransversal filter with four tappered delay lines combining the signalson the two principal states of polarization (PSP).
 12. Method inaccordance with claim 11 in which the gradient ∇H_(LP)(θ) with respectto θ is computed as: $\begin{matrix}{{\nabla{H_{lp}(\theta)}} = {4( {S_{1,l} - S_{1,p}} ){Re}\{ {\frac{1}{T_{u}}{\int_{t_{n}}^{t_{n + 1}}\lbrack {{{y_{1,l}^{*}(t)}{a_{l}^{T}(t)}} - {{y_{2,l}(t)}{b_{l}^{T}(t)}} -} }} }} \\{ { {{{y_{1,p}^{*}(t)}{a_{p}^{T}(t)}} + {{y_{2,p}^{*}(t)}{b_{p}^{T}(t)}}} \rbrack{\mathbb{d}t}\; J} \} +} \\{4( {S_{2,l} - S_{2,p}} ){Re}\{ {\frac{1}{T_{u}}{\int_{t_{n}}^{t_{n + 1}}\lbrack {{{y_{2,l}^{*}(t)}{a_{l}^{T}(t)}} + {{y_{1,l}(t)}{b_{l}^{T}(t)}} -} }} } \\{ { {{{y_{2,p}^{*}(t)}{a_{p}^{T}(t)}} - {{y_{1,p}^{*}(t)}{b_{j}^{T}(t)}}} \rbrack{\mathbb{d}t}\; J} \} -} \\{4( {S_{3,l} - S_{3,p}} ){Im}\{ {\frac{1}{T_{u}}{\int_{t_{n}}^{t_{n + 1}}\lbrack {{{y_{2,l}^{*}(t)}{a_{l}^{T}(t)}} + {{y_{1,l}(t)}{b_{l}^{T}(t)}} -} }} } \\ { {{{y_{2,p}^{*}(t)}{a_{p}^{T}(t)}} - {{y_{1,p}^{*}(t)}{b_{p}^{T}(t)}}} \rbrack{\mathbb{d}t}\; J} \}\end{matrix}$ where y_(1,l)(t), y_(2,l)(t) and y_(1,p)(t), y_(2,p)(t)are respectively the components y₁(t) e y₂(t) on the two orthogonalpolarizations of the compensator output signal filtered respectivelythrough a narrow band filter centered on the frequency f_(l) and f_(p);and a_(l)(t) e b_(l)(t) are the vectors: $\begin{matrix}{{a_{l}(t)} = \begin{pmatrix}{x_{1,l}(t)} \\{x_{1,l}( {t - {\alpha\;\tau_{c}}} )} \\{x_{1,l}( {t - \tau_{c}} )} \\{x_{1,l}( {t - \tau_{c} - {\alpha\tau}_{c}} )} \\{x_{2,l}(t)} \\{x_{2,l}( {t - {\alpha\;\tau_{c}}} )} \\{x_{2,l}( {t - \tau_{c}} )} \\{x_{2,l}( {t - \tau_{c} - {\alpha\;\tau_{c}}} )}\end{pmatrix}} & {{b_{l}(t)} = \begin{pmatrix}{x_{2,l}^{*}( {t - {2\tau_{c}}} )} \\{x_{2,l}^{*}( {t - \tau_{c} - {\beta\;\tau_{c}}} )} \\{x_{2,l}^{*}( {t - \tau_{c}} )} \\{x_{2,l}^{*}( {t - {\beta\;\tau_{c}}} )} \\{- {x_{1,l}^{*}( {t - {2\tau_{c}}} )}} \\{- {x_{1,l}^{*}( {t - \tau_{c} - {\beta\;\tau_{c}}} )}} \\{- {x_{1,l}^{*}( {t - \tau_{c}} )}} \\{- {x_{1,l}^{*}( {t - {\beta\;\tau_{c}}} )}}\end{pmatrix}}\end{matrix}$  with x_(1,l)(t) and x_(2,l)(t) which are respectivelysignals x₁(t) and x₂(t) on the two orthogonal polarizations of thecompensator input signal filtered with a narrow band filter centered onthe frequency f_(l) (similarly a_(l)(t) and b_(l)(t) for y_(1,p)(t) andy_(2,p)(t)) with the frequency f_(l)), and J is the Jacobean matrix ofthe transformation c=c(θ) defined as $\begin{matrix}{J\hat{=}\begin{pmatrix}\frac{\partial c_{1}}{\partial\phi_{1}} & \frac{\partial c_{1}}{\partial\phi_{2}} & \frac{\partial c_{1}}{\partial\theta_{1}} & \frac{\partial c_{1}}{\partial\theta_{2}} \\\frac{\partial c_{2}}{\partial\phi_{1}} & \frac{\partial c_{2}}{\partial\phi_{2}} & \frac{\partial c_{2}}{\partial\theta_{1}} & \frac{\partial c_{2}}{\partial\theta_{2}} \\\vdots & \vdots & \vdots & \vdots \\\frac{\partial c_{8}}{\partial\phi_{1}} & \frac{\partial c_{8}}{\partial\phi_{2}} & \frac{\partial c_{8}}{\partial\theta_{1}} & \frac{\partial c_{8}}{\partial\theta_{2}}\end{pmatrix}} & (18)\end{matrix}$  with c₁, . . . , c₈ which are the tap coefficients of thefour tappered delay lines.
 13. Method in accordance with claim 1 inwhich said optical devices comprise a polarization controller withcontrol angles φ1, φ2 and two optical rotators with rotation angles θ1and θ2 and said parameters comprise said control angles φ1, φ2 and saidrotation angles θ1 and θ2.
 14. PMD compensator in optical fibercommunication systems comprising: a cascade of adjustable opticaldevices over which passes an optical signal to be compensated, and anadjustment system configured to take components y1(t) and y2(t) on thetwo orthogonal polarizations from the compensator output signal signal,the adjustment system comprising a controller which is operable tocompute, on the basis of said components taken, the Stokes parametersS₀, S₁, S₂, S₃ in a number Q of different frequencies of the compensatoroutput signal and to emit control signals for parameters of at leastsome of said adjustable optical devices so as to make virtually constantthe Stokes parameters computed at the different frequencies. 15.Compensator in accordance with claim 14 characterized in that saidoptical devices comprise a polarization controller with control anglesφ1, φ2 and two optical rotators with rotation angles θ₁ and θ₂ and inwhich said parameters for which the controller is operable to emit thecontrol sianals consist of said control angles θ1, θ2 and said rotationangles θ₁ and θ₂.
 16. Compensator in accordance with claim 15characterized in that between the controller and an optical rotator andbetween optical rotators there are fibers which introduce apredetermined differential unit delay maintaining the polarization.